Math in CTE Lesson Plan Template - DOC

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					                          Math-in-CTE Lesson Plan
Lesson Title: Maximizing Profits                                Lesson # 7
Author(s):                         Phone Number(s):           E-mail Address(es):
  Skip Walker                       720-423-6100     
  Dana Starbuck
Occupational Area: Business
CTE Concept(s): Break Even Analysis
Math Concepts: Solving Algebraic Equations & Linear Programming
Lesson Objective:        After completion of this lesson, the student should be able to
                         compute low level breakeven points and complex breakeven points
                         using linear programming
Supplies Needed:         Linear Programming Paper
                         Linear Programming Worksheet
                         Worksheet Answer Key
                         Graphing Review Sheet (if needed)

             THE "7 ELEMENTS"                                   TEACHER NOTES
                                                               (and answer key)
1. Introduce the CTE lesson.                      Discuss the definition of:
Many manufacturers produce several                Maximize: To increase or make as
products, they need to make enough of each        great as possible or to find the largest
product to meet their customers need and          value of a function
best utilize their use of materials, labor, and
                                                  Linear Programming: A mathematical
                                                  technique used in economics; finds the
As a manufacturer you need to decide which        maximum or minimum of linear functions
products to produce that will maximize your       in many variables subject to constraints
profits. This is done through a process
                                                  Break-even Point: The point, especially
called Linear Programming
                                                  the level of sales of a good or service, at
Before we can discuss maximum profits, we         which the return on investment is exactly
must first find out when the manufacturer will    equal to the amount invested.
begin making a profit. This is called the
                                                  *Break-even point has previously been
Break-even Point

2. Assess students’ math awareness as it          Break-even Point: Income=Expenses
   relates to the CTE lesson.
                                                  Equation:     3x + 18,000 = 15x
To assess student’s math background, give a                     -3x            -3x
short warm-up quiz with questions relating to                         18,000 = 12x
Break-even.                                                            12      12
What is the formula for Break-even Point?                               X = 1500
Try this problem: We are making widgets:          That is, 1500 Widgets need to be sold to
variable costs $3 per widget; fixed costs total   make a profit.
$18,000; and we are selling them for $15
each. What would the breakeven point be?
3. Work through the math example embedded         Linear Programming Paper has been
   in the CTE lesson.                             included; it is nice to be able to do the
Otto Toyom builds toy cars and toy trucks.        problem at the top of the page, while
Each car needs 4 wheels, 2 seats, and 1 gas       having the graph paper included.
tank. Each truck needs 6 wheels, 1 seat, and      Constraint: The state of being restricted
3 gas tanks. His storeroom has 36 wheels,         or confined within prescribed bounds.
14 seats, and 15 gas tanks. He makes $1.00
on each car and $1.00 on each truck he            KEY:
sells.                                            2. Wheels [4x + 6y ≤ 36]
What combination of cars and trucks will          In English, this equation says that each
maximize Otto’s profit?                           car needs 4 wheels, each truck needs 6
Steps we must take to solve this problem:         wheels and together they can not add to
                                                  be more than 36 (which is the number of
1. Pick variables.                                wheels Otto can hold in his storeroom).
   Let x = # cars and y = # trucks.
                                                  Seats [2x + y ≤ 14]
2. Write inequalities that model each
   constraint.                                    Gas tanks [x + 3y ≤ 15]

3. Graph each constraint.                         3. Graphing inequalities will need to be
                                                  reviewed with students. (See attached
4. Calculate intersections of each inequality.    sheet for further review).
   (Using Systems of       Equations)
                                               Students will also need to understand
5. Write a Profit Equation.                    why we are looking at graphs only in the
6. Substitute each ordered pair from Step #4 first quadrant (where x and y are both
into the profit       equation from step #5.   positive). Explain that there is no way for
Determine which ordered pair has the           Otto to make a negative number of either
highest profit. This is the Linear Combination cars or trucks, thus x and y must both be
that maximizes the profit!                     positive.
                                                  4. Students must know that ANY
                                                  maximum or minimum will occur at one of
                                                  the vertices. (These are the intersections
                                                  of the inequalities). Solving Systems of
                                                  Equations will need to be reviewed with
                                                  students. (See attached for further
                                                  Intersections: (6, 2), (3, 4), (5.4, 3.2)
                                                  use (5, 3) since you can’t make a fraction
                                                  of a car or truck.
                                                  5. Profit = 1x + 1y
                                                  6. Max Profit:
                                                 (6,2) Profit = $8
                                                 (3, 4) Profit = $7
                                                 (5, 3) Profit = $8
                                                 So Max occurs either with 6 cars and 2
                                                 trucks or 5 cars and 3 trucks.
4. Work through related, contextual math-in-     KEY:
   CTE examples.
                                                 Profit = 1x + 2y
Truck drivers have just become popular
because of a new TV series called “Big Red       (6,2) Profit = $10
Ed.” Toy trucks are a hot item. Otto can now     (3, 4) Profit = $11
make $2.00 per truck though he stills gets
                                                 (5, 3) Profit = $11
$1.00 per car. He has hired you as a
consultant to advise him, and your salary is a   So now, the Max occurs either with 3 cars
percentage of the total profits. What is his     and 4 trucks or 5 cars and 3 trucks.
best choice for the number of cars and the
number of trucks to make now? How can you
be sure? Explain.
5. Work through traditional math examples.     USE LINEAR PROGRAMMING PAPER
The Smiths have a small farm where they        FOR WORK.
grow corn and tomatoes for sale. They have     1. Let x = the number of acres of corn
a total of 21 acres available for planting.    planted and y = the number of acres of
Because they cannot afford to pay a lot for    tomatoes planted.
help, they have many restrictions based on     2. Write inequalities which represent the
their labor. They have a total of 150 hours    restrictions on the land available. (x + y ≤
available for planting and 130 hours available 21)
for harvesting. Each acre of corn takes 6      The time available for planting:
hours to plant and 4 hours to harvest. Each    (6x + 10y ≤ 150)
acre of tomatoes takes 10 hours to plant and The time available for harvesting:
10 hours to harvest. A local grocery chain     (4x + 10y ≤ 130)
has agreed to purchase 3 acres worth of        The arrangement made with the grocery
tomatoes.                                      chain: (y ≥ 3)
If each acre of corn can be sold for $600 and    3-4.Graph the system you found in part
each acre of tomatoes can be sold for $800,      (a) and find the vertices: (Vertices (x ,
how many acres of each type should the           y): (0, 3); (18, 3): (15, 6); (10, 9); (0, 13))
Smiths plant?                                    5. Profit = 600x + 800y
                                                 6. Substitute each vertex into the Profit
                                                 (0,3) Profit = $2,400
                                                 (18,3) Profit = $13,200
                                                 **MAX**(15,6) Profit = $13,800
                                                 (10,9) Profit = $13,200
                                                 (0, 13) Profit = $10,400
6. Students demonstrate their understanding.    See Answer Sheets for completed
See Attached Linear Programming                 Graphs.
Worksheet Use with Linear Programming           ANSWERS:
Paper (This assignment may take several
nights as the problems can be challenging)
                                                a. Profit=30T + 15C
                                                b. T + C ≤ 40000; T + C ≥ 18000; 6C +
                                                   2T ≤ 120000
                                                c. See graph
                                                d. (Texas, Calif.) = (40,000, 0)
                                                Profit = $1200000
                                                a. x=old member; y=new member
                                                    Profit =10x + 8 y
                                                b. x ≥ 0, y ≥ 0; x ≤ 9, y ≤ 8; 6 ≤ x + y ≤
                                                   15, y ≥ 3; y ≥ x, y ≤ 3x
                                                c. See graph
                                                d. No, you must have old members
                                                   because the graph isn’t shaded at x =
                                                e. (old, new) = (9,6) Profit = $138
                                                f. (old, new) = (1.5, 4.5) but since we
                                                   can’t have 0.5 of a person, use (2, 4)
                                                   Profit = $52
                                                g. Profit=10x + 12y (old, new) = (7, 8)
                                                   Profit= $166
                                                a. x = Camel, y = Hippo
                                                   Profit = 300x + 200y
                                                b. x + y ≤ 12; x ≤ 11, y ≤ 7; y ≤ 2x;
                                                   100x + 200y ≥ 1000
                                                c. See graph
                                                (Camel, Hippo) = (11, 1) Profit = $3500
7. Formal assessment.                           ANSWER:
You are working at a small art store and your   X = Pastel Y= Watercolor
boss has decided to allow you sell some of     Profit = 40x + 100y
your art. Your boss has determined that you    Constraints: 5x + 15y ≤ 180; x + y ≤ 16;
can choose what combination of art you         x ≥ 3, y ≥ 2
would like to sell, so that you can make the
most profit possible.                          Possible Maximum’s: (3, 11); (3, 2); (6,
                                               10); (14, 2)
1. Each pastel requires $5 in materials and
   earns a profit of $40.
2. Each watercolor requires $15 in materials   MAX: (6, 10) Profit =$1240
   and earns a profit of $100.
3. You have $180 to spend on materials.
4. You plan to make at least 3 pastels and
   at least 2 watercolors.
5. You can make at most 16 pictures.
What is the optimum number of pastels and
watercolors that produces the maximum

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